How to Find the Period of a Function?

The time interval between two waves is known as a period, while a function that repeats its values at regular intervals or periods is known as a periodic function. In other words, the periodic function is a function that repeats its values after each particular period.

Step by step guide to a periodic function

A function \(y = f (x)\) is a periodic function in which there exists a positive real number \(P\) such that \(f (x + P) = f (x)\), for all \(x\) belong to real numbers. The smallest value of a positive real number \(P\) is called the fundamental period of a function. This fundamental period of a function is also called the function period in which the function repeats itself.

\(\color{blue}{f(x+P)=f(x)}\)

Note: the sine function is a periodic function with a period of \(2π\). \(sin(2π + x) = sinx\).

Below are diagrams of some of the periodic functions. The graph of each of the following periodic functions has translational symmetry.

Periods of some important periodic functions

The period of a function helps us to know the interval, after which the range of the periodic function is repeated. The domain of a periodic function \(f(x)\) includes the real number values of \(x\), and the range of a periodic function is a limited set of values within an interval. The length of this repeating interval, or the interval after which the range of the function repeats itself, is called the period of the periodic function.

The periods of some important periodic functions are as follows:

• The period of \(sinx\) and \(cosx\) is \(2π\).
• The period of \(tanx\) and \(cotx\) is \(π\).
• The period of \(secx\) and \(cosecx\) is \(2\).

Properties of periodic functions

The following features are useful for a deeper understanding of the concepts of periodic function:

• The graph of a periodic function is symmetric and repeats itself along the horizontal axis.
• The domain of the periodic function includes all values of real numbers, and the range of the periodic function is defined for a fixed interval.
• The period of a periodic function against which the period is repeated is equal to the constant over the whole range of the function.
• If \(f (x)\) is a periodic function with period \(P\), \(\frac{1}{f(x)}\) will also be a periodic function with the same fundamental period \(P\).
• If \(f(x)\) is a periodic function with a period of \(P\), then \(f(ax + b)\) is also a periodic function with a period of \(\frac {P}{|a|}\).
• If \(f(x)\) is a periodic function with a period of \(P\), then \(af(x) + b\) is also a periodic function with a period of \(P\).

Periodic Function – Example 1:

Find the period of the periodic function \(y=sin(4x + 5)\).

Solution:

The period of \(sinx\) is \(2π\), and the period of \(sin(4x + 5)\) is :

\(\frac{2π}{4}=\frac{π}{2}\)

Therefore, the period of \(sin(4x + 5)\) is \(\frac{π}{2}\).

Periodic Function – Example 2:

Find the period of the periodic function \(y=9 cos(6x + 4)\).

The period of \(cosx\) is \(2π\), and the period of \(9 cos(6x + 4)\) is:

\(\frac{2π}{6}=\frac{π}{3}\)

Therefore, the period of \(9 cos(6x + 4)\) is \(\frac{π}{3}\).

Exercises for Periodic Function

Find the period of the function.

1. \(\color{blue}{y= tan3x + sin\frac{5x}{2}}\)
2. \(\color{blue}{y=sec(\pi x-2)}\)
3. \(\color{blue}{y=cot(-(\frac{2\pi}{3})x)}\)
4. \(\color{blue}{\:y=cos\left(-\left(\frac{2}{3}\right)x-\pi \right)}\)

1. \(\color{blue}{4\pi}\)
2. \(\color{blue}{2}\)
3. \(\color{blue}{\frac{3}{2}}\)
4. \(\color{blue}{3\pi}\)